Less than two weeks before Donald J. Trump is set to go on trial on criminal charges in Manhattan, the prosecutors who brought the case proposed a delay of up to 30 days, a startling development in the first prosecution of a former American president.

The Manhattan district attorney’s office, which accused Mr. Trump of covering up a sex scandal during and after the 2016 presidential campaign, said the delay would give Mr. Trump’s lawyers time to review a new batch of records. The office sought the records more than a year ago, but only recently received them from federal prosecutors, who years ago investigated the hush-money payments at the center of the case.

In response to the records — tens of thousands of pages of them — Mr. Trump’s lawyers requested that the trial be delayed 90 days. Although the former president frequently requests such delays, prosecutors consenting to any postponement makes one far more likely.

Mr. Trump, who clinched the Republican presidential nomination for the third time this week, faces four criminal trials and several civil lawsuits. The Manhattan case had been the only one of the four criminal cases not mired in delays.

Now it too appears likely to be postponed, though it remains on track to reach trial before Election Day.